Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Generalized jordan curve theorem

  1. May 2, 2007 #1

    StatusX

    User Avatar
    Homework Helper

    Is anyone here familiar with the proof (using homology) of the generalized Jordan curve theorem, that a subspace of S^n homeomorphic to S^(n-1) divides it into two components? It can be found on page 169 of Hatcher's algebraic topology book, which can be downloaded from this page.

    I'm wondering if it's possible to use the same kind of proof to show that a general (n-1)-manifold divides S^n into two components. It was shown that the complement of an n-1-sphere in S^n actually has the homology of S^0, which is much stronger, and won't hold for general n-1-manifolds. But all I need is that the 0th reduced homology group is Z, which does seem to be true. The problem is that the corresponding terms of the Mayer-vietoris sequence used in Hatcher's proof aren't zero where they need to be. Does anyone have any ideas?
     
  2. jcsd
  3. May 2, 2007 #2

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

    are you using alexander duality? see p.254.
     
    Last edited: May 2, 2007
  4. May 2, 2007 #3

    StatusX

    User Avatar
    Homework Helper

    I havent gotten to that yet (Im just starting cohomology), though I see how it would follow from there. Is there an easier method, or a way to extract that bit of information using the idea of alexander duality without developing all the machinery?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Generalized jordan curve theorem
  1. Jordan Product (Replies: 1)

Loading...